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BigNum in pure javascript

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npm install --save bn.js


const BN = require('bn.js');

var a = new BN('dead', 16);
var b = new BN('101010', 2);

var res = a.add(b);
console.log(res.toString(10));  // 57047

Note: decimals are not supported in this library.


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There are several prefixes to instructions that affect the way the work. Here is the list of them in the order of appearance in the function name:

  • i - perform operation in-place, storing the result in the host object (on which the method was invoked). Might be used to avoid number allocation costs
  • u - unsigned, ignore the sign of operands when performing operation, or always return positive value. Second case applies to reduction operations like mod(). In such cases if the result will be negative - modulo will be added to the result to make it positive


  • n - the argument of the function must be a plain JavaScript Number. Decimals are not supported.
  • rn - both argument and return value of the function are plain JavaScript Numbers. Decimals are not supported.


  • a.iadd(b) - perform addition on a and b, storing the result in a
  • a.umod(b) - reduce a modulo b, returning positive value
  • a.iushln(13) - shift bits of a left by 13


Prefixes/postfixes are put in parens at the of the line. endian - could be either le (little-endian) or be (big-endian).


  • a.clone() - clone number
  • a.toString(base, length) - convert to base-string and pad with zeroes
  • a.toNumber() - convert to Javascript Number (limited to 53 bits)
  • a.toJSON() - convert to JSON compatible hex string (alias of toString(16))
  • a.toArray(endian, length) - convert to byte Array, and optionally zero pad to length, throwing if already exceeding
  • a.toArrayLike(type, endian, length) - convert to an instance of type, which must behave like an Array
  • a.toBuffer(endian, length) - convert to Node.js Buffer (if available). For compatibility with browserify and similar tools, use this instead: a.toArrayLike(Buffer, endian, length)
  • a.bitLength() - get number of bits occupied
  • a.zeroBits() - return number of less-significant consequent zero bits (example: 1010000 has 4 zero bits)
  • a.byteLength() - return number of bytes occupied
  • a.isNeg() - true if the number is negative
  • a.isEven() - no comments
  • a.isOdd() - no comments
  • a.isZero() - no comments
  • a.cmp(b) - compare numbers and return -1 (a < b), 0 (a == b), or 1 (a > b) depending on the comparison result (ucmp, cmpn)
  • - a less than b (n)
  • a.lte(b) - a less than or equals b (n)
  • - a greater than b (n)
  • a.gte(b) - a greater than or equals b (n)
  • a.eq(b) - a equals b (n)
  • a.toTwos(width) - convert to two's complement representation, where width is bit width
  • a.fromTwos(width) - convert from two's complement representation, where width is the bit width
  • BN.isBN(object) - returns true if the supplied object is a BN.js instance
  • BN.max(a, b) - return a if a bigger than b
  • BN.min(a, b) - return a if a less than b


  • a.neg() - negate sign (i)
  • a.abs() - absolute value (i)
  • a.add(b) - addition (i, n, in)
  • a.sub(b) - subtraction (i, n, in)
  • a.mul(b) - multiply (i, n, in)
  • a.sqr() - square (i)
  • a.pow(b) - raise a to the power of b
  • a.div(b) - divide (divn, idivn)
  • a.mod(b) - reduct (u, n) (but no umodn)
  • a.divmod(b) - quotient and modulus obtained by dividing
  • a.divRound(b) - rounded division

Bit operations

  • a.or(b) - or (i, u, iu)
  • a.and(b) - and (i, u, iu, andln) (NOTE: andln is going to be replaced with andn in future)
  • a.xor(b) - xor (i, u, iu)
  • a.setn(b, value) - set specified bit to value
  • a.shln(b) - shift left (i, u, iu)
  • a.shrn(b) - shift right (i, u, iu)
  • a.testn(b) - test if specified bit is set
  • a.maskn(b) - clear bits with indexes higher or equal to b (i)
  • a.bincn(b) - add 1 << b to the number
  • a.notn(w) - not (for the width specified by w) (i)


  • a.gcd(b) - GCD
  • a.egcd(b) - Extended GCD results ({ a: ..., b: ..., gcd: ... })
  • a.invm(b) - inverse a modulo b

Fast reduction

When doing lots of reductions using the same modulo, it might be beneficial to use some tricks: like Montgomery multiplication, or using special algorithm for Mersenne Prime.

Reduction context

To enable this tricks one should create a reduction context:

var red =;

where num is just a BN instance.


var red =;

Where primeName is either of these Mersenne Primes:

  • 'k256'
  • 'p224'
  • 'p192'
  • 'p25519'


var red = BN.mont(num);

To reduce numbers with Montgomery trick. .mont() is generally faster than .red(num), but slower than

Converting numbers

Before performing anything in reduction context - numbers should be converted to it. Usually, this means that one should:

  • Convert inputs to reducted ones
  • Operate on them in reduction context
  • Convert outputs back from the reduction context

Here is how one may convert numbers to red:

var redA = a.toRed(red);

Where red is a reduction context created using instructions above

Here is how to convert them back:

var a = redA.fromRed();

Red instructions

Most of the instructions from the very start of this readme have their counterparts in red context:

  • a.redAdd(b), a.redIAdd(b)
  • a.redSub(b), a.redISub(b)
  • a.redShl(num)
  • a.redMul(b), a.redIMul(b)
  • a.redSqr(), a.redISqr()
  • a.redSqrt() - square root modulo reduction context's prime
  • a.redInvm() - modular inverse of the number
  • a.redNeg()
  • a.redPow(b) - modular exponentiation

Number Size

Optimized for elliptic curves that work with 256-bit numbers. There is no limitation on the size of the numbers.


This software is licensed under the MIT License.